Intersection cohomology of moduli spaces of sheaves on surfaces

Manschot, Jan; Mozgovoy, Sergey

doi:10.1007/s00029-018-0431-1

Cited by 15 publications

(13 citation statements)

References 37 publications

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“…The application of the Donaldson-Thomas formalism of [Mei15] to Gieseker semistable sheaves on del Pezzo surfaces is discussed in Example 3.34 of [Mei15] and in [MM18]. Lemma 3.6 will able us to apply this formalism to Bridgeland semistable objects in D b (P 2 ).…”

Section: Remarkmentioning

confidence: 99%

“…The key point is the relation between intersection cohomology and Donaldson-Thomas invariants, which goes back to [MR17]. This relation has been extended to Gieseker semistable sheaves on surfaces with negative canonical line bundles in [MM18] and to some abstract framework for categories of homological one in [Mei15].…”

Section: Introductionmentioning

confidence: 99%

“…In general, under the extra assumption r > 0, a different proof of Theorem 0.2 could be extracted from [MM18].…”

Section: Introductionmentioning

confidence: 99%

“…Manschot. An alternative algorithm to compute Hodge numbers of intersection cohomology of moduli spaces of Gieseker semistable sheaves on P 2 , at least for sheaves of positive rank, has been developed by Manschot [Man11][Man13][Man17] [MM18]. The idea, going back to Yoshioka [Yos94][Yos95] in rank 2, is to blow-up one point on P 2 , reduce the problem to the blown-up surface by some blow-up formula, and then solve the problem on the blown-up surface by wall-crossing in the space of polarizations of the blown-up surface.…”

Section: Introductionmentioning

confidence: 99%

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Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

Bousseau

2019

Preprint

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We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on P 2 . This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on P 2 . Contents0. Introduction 0.1. Description of S(D in u,v ) and D P 2 u,v 0.2. Structure of the proof of Theorem 0.1 0.3. Applications to moduli spaces of Gieseker semistable sheaves 0.4. Relations with previous and future works 0.5. Plan of the paper 0.6. Acknowledgments 1. The scattering diagram S(D in u,v ) 1.1. Local scattering diagrams 1.2. Scattering diagrams 1.3. The scattering diagrams S(D in ) 1.4. The scattering diagrams S(D in u,v ), S(D in q ± ) and S(D in cl ± ) 1.5. Action of ψ(1) on scattering diagrams 2. The scattering diagram D P 2 u,v 2.1. Coherent sheaves on P 2 2.2. Stability conditions 2.3. Moduli spaces and walls 2.4. Intersection cohomology invariants 2.5. Scattering diagrams from stability conditions 2.6. Action of ψ(1) on D P 2 u,v 3. Consistency of D P 2 u,v 3.1. Statement of Theorem 3.1 3.2. Local structure near a wall 3.3. Numerical invariants from mixed Hodge theory 3.4. Donaldson-Thomas formalism 3.5. Wall-crossing formula 3.6. End of the proof of Theorem 3.1

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In general, under the extra assumption r > 0, a different proof of Theorem 0.2 could be extracted from [MM18].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

Bousseau

2019

Preprint

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“…Following §5.4 and §5.5, we consider the quantum torus g = γ∈Γ Rx γ , R = K(St a /C), and define, for any special geometric stability condition σ = (Z σ , P σ ) ∈ assumption implies that the quantum torus restricted to degrees in Γ ∩ Z −1 σ (ℓ) is commutative. By the results of [51,50,56,48], we have…”

Section: Stability Conditions On Surfacesmentioning

confidence: 99%

Wall-crossing structures on surfaces

Mozgovoy¹

2022

Preprint

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Families of Bridgeland stability conditions induce families of stability data, wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. We show that geometric stability conditions on surfaces with nef anticanonical bundle have global dimension 2 and we study the resulting family of stability data. We formulate a conjecture relating this family to the family of stability data associated with a quiver with potential and we verify this conjecture for the projective plane. SERGEY MOZGOVOY MotivicHall algebra of an exact category 5.3. Wall-crossing in the Hall algebra 5.4. Wall-crossing in the quantum torus 5.5. Stability data associated to a stability condition 5.6. Families of stability data and stability conditions 6. Stability conditions on surfaces 6.1. Some conventions 6.2. Construction of stability conditions 6.3. Cone of positive planes 6.4. Geometric stability conditions 6.5. Gieseker and twisted stability 6.6. Large volume limit 6.7. Bogomolov-type inequalities 6.8. Alternative parametrization of stability conditions 7. Stability data associated to geometric stability conditions 7.1. Vanishing results 7.2. Geometric stability data 7.3. Relation to intersection cohomology 8. Relation to quivers with potentials 8.1. Exceptional collections and tilting objects 8.2. Local surfaces 8.3. Relation to quiver representations 8.4. Quivers with potential and their motivic invariants 8.5. Relating stability data 8.6. Gluing families of stability data References

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Vafa–Witten Theory and Iterated Integrals of Modular Forms

Manschot

2019

Commun. Math. Phys.

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Vafa-Witten (VW) theory is a topologically twisted version of N = 4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU (N ) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU (2) and is considered on the complex projective plane P 2 . This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU (3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU (3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU (3) partition function is mathematically an example of a mock modular form of depth two.

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Intersection cohomology of moduli spaces of sheaves on surfaces

Cited by 15 publications

References 37 publications

Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

Wall-crossing structures on surfaces

Vafa–Witten Theory and Iterated Integrals of Modular Forms

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